C₂×#848397 #72337902

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#848397

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2433	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

47261 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 30 points:

(17,22)(18,21)(23,24)(25,29)(26,30)(27,28);(7,8,10,12)(9,11,13,14)(15,16)(17,21)(18,22)(19,20);(3,4)(5,6)(9,13)(11,14)(16,20)(17,22)(23,25,28,26)(24,29,27,30);(4,5)(7,8)(9,14)(10,12)(11,13)(15,17,19,22)(16,21,20,18)(23,26,28,25)(24,29,27,30);(1,2);(3,5,6,4)(7,9)(8,11)(10,13)(12,14)(15,18,19,21)(16,17,20,22)(23,27,28,24)(25,30,26,29)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232360, C₂×#72232336
maximal normal subgroup of	C₂×#72232360, C₂×#72232336

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((17,22)(18,21)(23,24)(25,29)(26,30)(27,28), (7,8,10,12)(9,11,13,14)(15,16)(17,21)(18,22)(19,20), (3,4)(5,6)(9,13)(11,14)(16,20)(17,22)(23,25,28,26)(24,29,27,30), (4,5)(7,8)(9,14)(10,12)(11,13)(15,17,19,22)(16,21,20,18)(23,26,28,25)(24,29,27,30), (1,2), (3,5,6,4)(7,9)(8,11)(10,13)(12,14)(15,18,19,21)(16,17,20,22)(23,27,28,24)(25,30,26,29));

C2×#848397 #72337902